The main result of the present work is a refinement of Theorem REF  into a LDP both for a Nash equilibrium and the state processes at equilibrium.
It is well-known by the celebrated Varadhan-Bryc equivalence, given in [1]}, [2]} (see also [3]}) that the LDP is equivalent to the so-called Laplace principle which can be stated as follows:
Given a function \(\mathcal {I}:\mathcal {P}(E)\rightarrow [0,\infty ]\)  with (weakly) compact sublevel sets \(\lbrace \mu \in \mathcal {P}(E): \mathcal {I}(\mu ) \le a \rbrace \)  called a (good) rate function, a sequence of measures \((\mu ^N)_{N \in \mathbb {N}}\)  on the Polish space \(E\)  satisfies the Laplace principle (in the weak topology) if for every bounded continuous function \(F:\mathcal {P}(E) \rightarrow \mathbb {R}\)  it holds
\(\lim _{N\rightarrow \infty }-\frac{1}{N}\log (E\left[\exp (-NF(\mu ^N)) \right]) = \inf _{\mu \in \mathcal {P}(E)}\left(F(\mu ) - \mathcal {I}(\mu ) \right).\) 
